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Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?

Standard community wiki rules apply: one quote per post.

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Why did a question with so much positive feedback get closed? –  Romeo Nov 28 '10 at 23:21
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Diminishing marginal utility. –  Qiaochu Yuan Jan 31 '11 at 2:46
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Closing this solved what problem? –  Matt Brin Jan 18 '12 at 18:35
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@Matt: standards for what kind of questions people want on MO have changed over time, and keeping this question opens gives a false impression to new users of what kind of questions we want on MO. It's less confusing to close it. This happens on other SE sites as well; many of the most popular questions on StackOverflow, for example, are also closed. There's also the more practical issue that if it's open people keep adding answers and, again, the marginal utility of each additional answer is decreasing. –  Qiaochu Yuan Nov 12 '13 at 3:03

97 Answers 97

Grothendieck comparing two approaches, with the metaphor of opening a nut: the hammer and chisel approach, striking repeatedly until the nut opens, or just letting the nut open naturally by immersing it in some soft liquid and let time pass:

"I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance."

Grothendieck, of course, always pioneered this approach, and considered for example that Jean-Pierre Serre was a master of the "hammer and chisel" approach, but always solving problems in a very elegant way.

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A friend who I won't name at the moment once told me this, paraphrased: An excellent problem-solver might not always be a great mathematician, while a bad problem-solver can still be an okay mathematician. On the other hand, a good Grothendieck is a great mathematician, while a bad Grothendieck is really terrible! –  Greg Kuperberg Dec 18 '09 at 7:20
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Greg, the symmetry in that statement would be nicer if you replaced "problem-solver" with "Erdos." –  Qiaochu Yuan Dec 27 '09 at 8:11
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That's a good suggestion, but I can't edit comments. So let me just rephrase the aphorism: "An excellent Erdos might reach certain limits as a mathematician, while a bad Erdos can still be an okay mathematician. On the other hand, a good Grothendieck can be a truly great mathematician, while a bad Grothendieck is really terrible." –  Greg Kuperberg Jan 19 '10 at 19:36

"The question you raise, "how can such a formulation lead to computations?" doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered." - Grothendieck

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"There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and… modular forms." - Eichler

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Described as apocryphal in at least one source --- see other question. –  Greg Kuperberg Nov 29 '09 at 20:44

(Caveat for all of mine: I've not hunted down primary sources to check that they're properly attributed)

"Manifolds are a bit like pornography: hard to define, but you know one when you see one." -S. Weinberger

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"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." - Weyl

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I never understood this quote, I mean, there's nasty algebra and nice algebra, so isn't that statement a bit strong? –  Harry Gindi Nov 30 '09 at 6:28
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I actually think it refers to something Mephistophelian about algebra, as if such heights of abstraction are meant not for mortals, or such symbolic calculation lacks intuitive "soul". (That would be a tendentious way of putting it!) –  Todd Trimble Jan 31 '11 at 1:39
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Oh, and I swear that I wrote that before seeing Anton Petrunin's contribution! –  Todd Trimble Jan 31 '11 at 1:42

We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?" - Gian-Carlo Rota

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Depends on the writer? –  Harry Gindi Nov 30 '09 at 13:47
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Similarly, depends on the mathematician.... –  Suvrit Feb 1 '11 at 18:04

"The shortest path between two truths in the real domain passes through the complex domain." -- Jacques Hadamard

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Here's a simple example. What is the radius of convergence for the power series of 1/(x^2 + 1) centered at 0? Looking only at the real line there's no apparent reason for the radius to be only 1. But in the complex plane, you can see that the radius is 1 because that's the distance from the center to the singularity at i. Another example would be using contour integration to compute integrals over the real line. –  John D. Cook Mar 26 '10 at 18:35

"The art of doing mathematics is finding that special case that contains all the germs of generality." -- David Hilbert

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The purpose of computing is insight, not numbers.

— Richard Hamming (1962)

The attitude adopted in this book is that while we expect to get numbers out of the machine, we also expect to take action based on them, and, therefore we need to understand thoroughly what numbers may, or may not, mean. To cite the author's favorite motto,

“The purpose of computing is insight, not numbers,” although some people claim,

“The purpose of computing numbers is not yet in sight.”

There is an innate risk in computing because “to compute is to sample, and one then enters the domain of statistics with all its uncertainties.”

– Richard W. Hamming, Introduction to applied numerical analysis, McGraw-Hill 1971, p.31.

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My favorite math quote will probably always be Paul Gordan's response to Hilbert's proof of his Basis Theorem: "This is not Mathematics. This is Theology."

Along with his redaction after he came to accept the method: "I have convinced myself that even theology has its merits."

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To find out how misunderstood is the relationship between Gordan and Hilbert read McLarty's 'Theology and its discontents'. cwru.edu/artsci/phil/Theology%20and%20its%20discontents.pdf –  David Corfield Mar 31 '10 at 7:47

«Allez en avant, et la foi vous viendra.»

Free translation: keep going, faith will come later.

Jean-le-Rond D'Alembert, to his students (quoted by Florian Cajori in A history of mathematics)

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A mathematician is a device for turning coffee into theorems. —Alfréd Rényi, but often attributed also to Paul Erdős

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Ken Ribet once told me the story of how he was sent a freebie book "for possible use in your undergraduate classes" that he looked at and decided he didn't want, so took it to the second hand bookstore in his lunch break, sold it, and bought lunch with the proceeds. On the way back to the math department he realised he'd turned theorems into coffee. –  Kevin Buzzard Nov 29 '09 at 22:42
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A comathematician is a device for turning cotheorems into ffee. A cotheorem is of course what one deduces from a rollary. –  Saul Glasman Nov 30 '09 at 19:02

La vie est étrange. En fait, en géometrie, on ne se représente pas de la même manière une droite complexe affine (par exemple pour le théorème de Ceva dans un triangle) et le corps des complexes x+iy. Quand j'y songe, les points imaginaires de la géometrie sont gris, les points réels noirs, et l'intersection de deux droites imaginaires conjuguées est un point réel noir. La belle conique ombilicale est argentée, les droites et cônes isotropes sont plutôt roses.

Laurent Schwartz, Un mathématicien aux prises avec le siècle.

Free translation: «Life is strange. In fact, in geometry, we do not think in the same way of a complex affine line (for example in the theorem of Ceva in a triangle) and of the field of complex numbers x+iy. When I think about this, imaginary points in geomtry are gray, the real points are black, and the intersection of two conjugate imaginary lines is a black real point. The beautiful umbilical conic is silver, the lines and isotropic cones are mostly pink.»

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Reminds me of the synaesthetic experiences of Feynman: "When I see equations, I see the letters in colors – I don't know why. As I'm talking, I see vague pictures of Bessel functions from Jahnke and Emde's book, with light-tan j's, slightly violet-bluish n's, and dark brown x's flying around. And I wonder what the hell it must look like to the students." –  Todd Trimble Jun 9 '13 at 14:42

"God exists since mathematics is consistent, and the Devil exists since we cannot prove it."- André Weil

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So, mathematics is consistent? I think we can more or less rely on dividing by three, for the rest we shouldn't be that sure... –  Jose Brox Dec 8 '09 at 15:42
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@Jose: I don't know if you were referring to this paper indirectly, but you should check out math.dartmouth.edu/~doyle/docs/three/three.pdf, where it is proven that division by three is possible. –  Steven Gubkin Apr 23 '10 at 12:42

It's hard to beat John Stembridge's page of quotes. My single favorite one on this page: "If I have not seen as far as others, it is because there were giants standing on my shoulders." - Hal Abelson.

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Along those lines, R. W. Hamming said "Mathematicians stand on each other’s shoulders while computer scientists stand on each other’s toes." –  John D. Cook Nov 30 '09 at 3:02
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My version of this quote, after working my way backwards through a series of papers, each relying on the previous ones: If I can't see a darn thing it's because I stand on the shoulders of giants... –  Ehud Friedgut Dec 2 '09 at 5:49
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What was Gell-Mann's version? Something like, "If I have seen farther than others, it's because I'm surrounded by pygmies?" –  Todd Trimble Jan 31 '11 at 1:15

"Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is."- Paul Erdős

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+1 I cannot tell you how frequently I use this quote in explaining to others (or to avoid explaining to others) how wonderful mathematics is. –  Jesse Madnick Jul 11 '10 at 3:32

“The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.”- Godfrey Harold Hardy

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Dirichlet allein, nicht ich, nicht Cauchy, nicht Gauß, weiß, was ein vollkommen strenger Beweis ist, sondern wir lernen es erst von ihm. Wenn Gauß sagt, er habe etwas bewiesen, so ist es mir sehr wahrscheinlich, wenn Cauchy es sagt, ist ebensoviel pro als contra zu wetten, wenn Dirichlet es sagt, ist es gewiß; ich lasse mich auf diese Delikatessen lieber gar nicht ein.

C. G. J. Jacobi, writing to von Humboldt, in 1846.

Without pretty ßs: Only Dirichlet, Not I, not Cauchy, not Gauss, knows what a perfectly rigourous proof is, but we learn it only from him. When Gauss says he has proved something, I think it is very likely; when Cauchy says it, it is a fifty-fifty bet; when Dirichlet says it, it is certain; I prefer not to go into these delicate matters.

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Oh I never get tired of cribbing about German along with Mark Twain .. baetzler.de/humor/the_awful_german_language.var .. –  Anweshi Feb 6 '10 at 16:20

You know, for a mathematician, he did not have enough imagination. But he has become a poet and now he is fine.

D. Hilbert, talking about an ex-student. I'd love to remember where I got this from!

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You might have gotten it from the Monthly in 1993, or a mailing list or blog that derives from that source. Google Books traces it back to a book about Ernst Cassirer published in 1949. Since it is quoted there as Cassirer's verbal story, it's less clear what Hilbert himself said. archive.org/stream/philosophyoferns033109mbp/… –  Greg Kuperberg Nov 29 '09 at 22:14
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This story is mentioned in Constance Reid's lovely biography "Hilbert". –  Lea M May 16 '11 at 4:27

"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." - Henry Poincaré

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"You don't have to believe in God, but you should believe in The Book." --- Paul Erdős. describing the Book held by the God that contains the most beautiful proofs to all the theorems

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You wanted probably this: "You don't have to believe in the S.F., but you should believe in The Book." –  Péter Komjáth Nov 8 '10 at 13:41

"My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful."- Herman Weyl

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"A mathematician who is not also something of a poet will never be a perfect mathematician"- Karl Weierstraß

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"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions", F. Klein (from Reed & Simon: Methods of modern mathematical physics)

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Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.

Gian-Carlo Rota, in an interview with David Sharp.

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Here is an online copy: fas.org/sgp/othergov/doe/lanl/pubs/00326965.pdf –  Jose Capco Nov 30 '09 at 11:11
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Combinatorics is discrete functional analysis in my world view, while functional analysis is applied combinatorics. –  Bill Johnson Mar 21 '10 at 19:28
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Bill, would you mind elaborating? As someone not particularly familiar with either field, I can imagine that by combinatorics being "discrete functional analysis" you mean e.g. generating function methods, or perhaps the general ambition of associating some sort of linear operator to combinatorial objects (e.g. adjacency matrix). But what do you mean by functional analysis being applied combinatorics. –  Erik Davis Apr 15 '10 at 1:01
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It takes balls to do combinatorics. –  Todd Trimble Jan 31 '11 at 1:45

Someone once told me that Grothendieck said "a sheaf of groups is a group of sheaves," although I have been unable to find a real reference. Can anyone substantiate this?

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This is the best way to think of a sheaf of groups though (IMO)- as a group object in the category of sheaves of sets. –  Steven Gubkin Nov 29 '09 at 21:38

" Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulation of numbers, the manipulation of structures." And if I had asked what music means to you, would you have answered: "The manipulation of notes?" "- Serge Lang

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"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."

Pierre de Fermat

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"A mathematical truth is neither simple nor complicated in itself, it is." - Émile Lemoine

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In mathematics you don't understand things. You just get used to them.

--John von Neumann, reply to a physicist at Los Alamos who had said "I don't understand the method of characteristics."

---- footnote on page 226 of Gary Zukav, The Dancing Wu Li Masters: An Overview of the New Physics, Rider, London, 1990.

(taken from Warren Dicks' Home Page)

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I don't know how much I agree with that. It may be true about physics, but not math. –  Harry Gindi Nov 30 '09 at 6:20
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I feel that it's true of both mathematics and physics, but when talking about mathematics it's a much deeper statement. –  Dan Piponi Dec 2 '09 at 19:37
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I may have a surprise for you... "Understanding"="getting used to"! :) –  efq Dec 4 '09 at 21:20
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I constantly try to check myself to make sure that I am really understanding, and am not just "getting used to" the things that I am learning. It is very difficult, but I think that I am gaining understanding of the basics little by little. –  Steven Gubkin Dec 7 '09 at 2:04
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I totally disagree with this quote ... –  Martin Brandenburg Feb 2 '10 at 15:45

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